Crooked Maps in Finite Fields

Gohar Kyureghyan

Abstract

We consider the maps f:2n →2n with the property that the set { f(x+a)+ f(x): x ∈F2n} is a hyperplane or a complement of hyperplane for every a ∈2n*. The main goal of the talk is to show that almost all maps f(x) = Σb ∈Bcb(x+b)d, where B ⊂2n and Σb ∈Bcb ≠0, are not of that type. In particular, the only such power maps have exponents 2i+2j with gcd(n, i-j)=1. We give also a geometrical characterization of this maps.